From 3b27994444b0370a5f0d465bb64c8d8ae242d041 Mon Sep 17 00:00:00 2001 From: Emanuele Paolini Date: Sun, 21 Feb 2021 18:58:58 +0100 Subject: [PATCH] rilascio (+hotfix) * disegno Darboux * disuguaglianze di Young e Hoelder * definizione asintoti * fix vari --- chapters/AnalisiUno-05-derivate.tex | 13 ++++++++++--- figures/darboux2.pdf | Bin 0 -> 70429 bytes 2 files changed, 10 insertions(+), 3 deletions(-) create mode 100644 figures/darboux2.pdf diff --git a/chapters/AnalisiUno-05-derivate.tex b/chapters/AnalisiUno-05-derivate.tex index ac77e28..9e161f0 100644 --- a/chapters/AnalisiUno-05-derivate.tex +++ b/chapters/AnalisiUno-05-derivate.tex @@ -433,7 +433,7 @@ \section{derivata} \end{align*} \end{proof} -\section nomenclatura +\section{nomenclatura} In questa sezione introdurremo una terminologia che è largamente utilizzata nello studio di funzione. @@ -501,6 +501,9 @@ \section{derivata} \lim_{x\to +\infty} f(x) - (mx+q) = 0. \] Se $m=0$ diremo che il grafico di $f$ ha un \emph{asintoto orizzontale} $y=q$ +\mymargin{asintoto orizzontale/obliquo}% +\index{asintoto!orizzontale}% +\index{asintoto!obliquo}% altrimenti diremo che $y=mx+q$ è un \emph{asintoto obliquo}. Stessa cosa si può dire per $x\to -\infty$. @@ -508,7 +511,7 @@ \section{derivata} \[ \lim_{x\to x_0} \abs{f(x)} = +\infty \] -diremo che la retta $x=x_0$ è un \myemph{asintoto verticale} per il +diremo che la retta $x=x_0$ è un \myemph{asintoto!verticale} per il grafico della funzione $f$. \end{definition} @@ -1063,6 +1066,7 @@ \section{criteri di monotonia} \begin{figure} \myurl{darboux}{Funzione con derivata non continua esempio \getrefnumber{ex:derivata_non_continua}}% \centering\includegraphics[height=3cm]{darboux.pdf} + \centering\includegraphics[width=\textwidth]{darboux2.pdf} \begin{comment} \begin{tikzpicture}[scale=5] \draw[->] (-0.5, 0) -- (0.5, 0) node[above] {$x$}; @@ -1080,8 +1084,11 @@ \section{criteri di monotonia} \end{comment} \caption{Il grafico di una funzione derivabile ma con derivata non continua (vedi esempio~\ref{ex:derivata_non_continua}). + L'ingrandimento nel disegno in basso rende evidente + il fatto che la derivata oscilla tra i valori + $-1$ e $1$ in ogni intorno di $0$. La proprietà di Darboux (teorema~\ref{th:darboux}) - rimane soddisfatta: la derivata assume tutti i valori + rimane comunque soddisfatta: la derivata assume tutti i valori compresi tra $1$ e $-1$ in ogni intorno di $x=0$ ma non ha limite per $x\to 0$. } diff --git a/figures/darboux2.pdf b/figures/darboux2.pdf new file mode 100644 index 0000000000000000000000000000000000000000..df330c75ffc844098a6a094500f3985d72f66f9e GIT binary patch literal 70429 zcmcJ&-L56sb*y*4pJLw#M}ncMz5drS2m)eLJV&q$5@Z85Ea;+1mUxKf9ef#H^_g{SX?MLqZ;q`3mAOHB}!?(ZH zOLU!=y}bX&58wXo<9~g<-p+@|3no8&{QkpNe|UL&cYJ&O-(TKd|MSb?_4hAFJ(tUO z7b9<6zP(=F-kr~f*FU_z91d@K^Mby4`P;+ka(H*UUtaI>=1o=`cE|(1dHEH$zB}I! zPtOW)^uuj(dG|OznGCR*9N*rZuE*U|3hYeI@9uA>y~zOA^`f7A`SSi}hu6c~>zDr` z$N1PU|B*8I-OJJf$b@m%e(viwtGr} z&E!gnANM8&b|&X{x9ffPle#fVNQY**S%8epF?;l-Yd;5(> z`&f;Z7Fu94xl`n)y-9(c$@Sgiwr`&I0XCBdMSj>f&jMSmJSg(ZlgR+vQx0$R%E!K; zEo5&*A)lU&2H6`u-W_iH#&&ocWM_1s1P}Y>b|~^WwZkYxEBmB&POTjeRhQ?7QyW5d 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